Kaluba, Marek
Nowak, Piotr W.
Ozawa, Narutaka
2018-01-26
<p>This is the dataset accompanying <em>Aut(š½ā
) has property (T)</em> paper (https://arxiv.org/abs/1712.07167). See Section 4 thereof for a detailed description of the content of the included files:</p>
<blockquote>
<p><code>tar --list -f ./oSAutF5_r2.tar.xz</code></p>
<p><br>
<code>oSAutF5_r2/<br>
oSAutF5_r2/1.3/<br>
oSAutF5_r2/1.3/full_2018-01-26T12:29:58.143.log<br>
oSAutF5_r2/1.3/solver_2018-01-26T12:29:58.143.log<br>
oSAutF5_r2/1.3/SDPmatrix.jld<br>
oSAutF5_r2/1.3/lambda.jld<br>
oSAutF5_r2/U_pis.jld<br>
oSAutF5_r2/pm.jld<br>
oSAutF5_r2/delta.jld<br>
oSAutF5_r2/orbits.jld<br>
oSAutF5_r2/preps.jld</code></p>
</blockquote>
<p>To replicate the computation of the spectral gap clone <code>1712.07167</code> repository first</p>
<blockquote>
<p><code>git clone https://git.wmi.amu.edu.pl/kalmar/1712.07167.git</code></p>
</blockquote>
<p>Then unpack the content of <code>oSAutF5_r2.tar.xz</code> into <code>1712.07167</code> folder.</p>
<p>You need <code>julia-1.1.0</code> or above. In <code>julia</code>s REPL run</p>
<blockquote>
<p><code>using Pkg<br>
Pkg.activate("1712.07167")<br>
Pkg.instantiate()<br>
Pkg.test("PropertyT")</code></p>
</blockquote>
<p>Finally, to verify that the Laplace operator on <em>SAut(š½ā
)</em> (associated to the standard generating set) has spectral gap of at least <code>1.3</code> run from within <code>1712.07167</code> folder</p>
<blockquote>
<p><code>julia check_SAutF5.jl</code></p>
</blockquote>
<p>If You want to generate the multiplication table and other files on Your own delete all <code>*.jld</code> files from the <code>oSAutF5_r2</code> folder (but the ones in <code>1.3</code> folder) and run the same command again. Note: You need at least <code>20</code>GB of RAM and spare a few hours of Your CPU.</p>
<p>We reproduce the content of <code>check_SAutF5.jl</code> script below.</p>
<blockquote>
<p><code>using Pkg<br>
Pkg.activate(".")<br>
using Groups<br>
using GroupRings<br>
using PropertyT<br>
using SparseArrays<br>
using LinearAlgebra<br>
using IntervalArithmetic<br>
using JLD</code></p>
<p><code>@show Threads.nthreads()<br>
BLAS.set_num_threads(Threads.nthreads());</code></p>
<p><code>G = SAut(FreeGroup(5))<br>
pm = load("oSAutF5_r2/pm.jld", "pm");<br>
RG = GroupRing(G, pm)<br>
@info RG</code></p>
<p><code>S_size = 80<br>
# due to technical problems we are no longer able to load delta.jl on julia-1.0<br>
Δ_coeff = SparseVector(maximum(pm), collect(1:(1+S_size)), [S_size; -ones(S_size)])<br>
Δ = GroupRingElem(Δ_coeff, RG);<br>
Δ² = Δ^2;</code></p>
<p><code>@info "Loading solution"<br>
λā = load("oSAutF5_r2/1.3/lambda.jld", "λ")<br>
Pā = load("oSAutF5_r2/1.3/SDPmatrix.jld", "P");</code></p>
<p><code>@info "Taking square root of P"<br>
@time Q = real(sqrt(Pā));</code></p>
<p><code>Q_aug, check_columns_augmentation = PropertyT.augIdproj(Interval, Q);<br>
@show check_columns_augmentation<br>
if !check_columns_augmentation<br>
@warn "Columns of Q are not guaranteed to represent elements of the augmentation ideal!"<br>
end</code></p>
<p><code>@info "Computing SOS decomposition"<br>
@time sos = PropertyT.compute_SOS(RG, Q_aug);</code></p>
<p><code>residual = Δ² - @interval(λā)*Δ - sos;<br>
@show norm(residual, 1)</code></p>
</blockquote>
<p> </p>
<p>This research was supported in part by</p>
<ul>
<li>PL-Grid Infrastructure,</li>
<li>grant 2015/19/B/ST1/01458, National Science Center, Poland</li>
<li>grant 2017/26/D/ST1/00103, National Science Center, Poland.</li>
</ul>
https://doi.org/10.5281/zenodo.1913734
oai:zenodo.org:1913734
Zenodo
https://arxiv.org/abs/arXiv:1712.07167
https://zenodo.org/communities/eu
https://doi.org/10.5281/zenodo.1133440
info:eu-repo/semantics/openAccess
Creative Commons Attribution Share Alike 4.0 International
https://creativecommons.org/licenses/by-sa/4.0/legalcode
Aut(F_5)
property (T)
Laplace operator
spectral gap
sum of squares
semidefinite optimization
An approximation of the spectral gap for the Laplace operator on SAut(Fā
)
info:eu-repo/semantics/other